On the equation 1^k + 2^k + ⋯ + x^k = yⁿ for fixed x

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• 作者：A. Bé ; rczes^a ; ^{berczesa@science.unideb.hu" class="auth_mail" title="E-mail the corresponding author} ; L. Hajdu^a ; ^{hajdul@science.unideb.hu" class="auth_mail" title="E-mail the corresponding author} ; T. Miyazaki^b ; ^{tmiyazaki@gunma-u.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; I. Pink^a ; ^c ; ^{pinki@science.unideb.hu" class="auth_mail" title="E-mail the corresponding author} ; ^{istvan.pink@sbg.ac.at" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11D61 ; 11D41
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：163
• 期：Complete
• 页码：43-60
• 全文大小：374 K

文摘

We provide all solutions of the title equation in positive integers x,k,y,n$x,k,y,n$ with 1≤x<25$1\le x<25$ and 11d" title="Click to view the MathML source">n≥3$n\ge 3$. For these values of the parameters, our result gives an affirmative answer to a related, classical conjecture of Schäffer. In our proofs we combine several tools: Baker's method (in particular, sharp bounds for the linear combinations of logarithms of two algebraic numbers), polynomial-exponential congruences and computational methods.